Resistors and Conductivity

A resistor is a component of a circuit that acts to reduce both the flow of current and the voltage levels within the circuit. When current runs through a resistor, the energy stored within particles is converted to another form of energy, typically indicated by the emission of light or heat. Conductivity is a property of a given material that refers to the material's ability to transmit electricity. Conductivity and resistivity are opposites; that is, the higher the conductivity of a material, the less resistance it offers to the flow of current.

Relevant Equations

The resistance of a material can be calculated in several ways. The most common method relates resistance to the potential difference and the conventional current of the circuit, using the equation [math]\displaystyle{ R = {\frac{ΔV}{I}} }[/math] where ΔV is the potential difference across the resistor and I is the conventional current running through the circuit.

Another equation used to quantify resistance relates it to certain properties of the material and geometric properties of the resistor itself: [math]\displaystyle{ R = {\frac{L}{σA}} }[/math] where L is the length of the resistor, σ is the conductivity of the material, and A is the cross-sectional area of the resistor. This equation clearly demonstrates that resistivity and conductivity are inverses, as the conductivity constant can be found in the denominator.

A Computational Model

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Symbol

The conventional symbol for a resistor used in electrical circuit diagrams is shown below.

Resistors in Series

When resistors are connected along a single path with no branches, they are said to be in series. Resistors in series are, in practice, equivalent to a single resistor with the combined resistance of its constituent resistors. In other words, [math]\displaystyle{ R_{equivalent} = R_1 + R_2 + R_3 + ... + R_n }[/math] for n resistors in series.

Because R = L/(σA), if every resistor is composed of the same material and has the same cross-sectional area, [math]\displaystyle{ L_{equivalent} = L_1 + L_2 + L_3 + ... + L_n }[/math]

Resistors in Parallel

When resistors are not connected in series, they can be connected in parallel, creating several branches within a circuit. Several resistors in parallel are, in practice, equivalent to a single resistor with a resistance that is the reciprocal of the sum of reciprocals of the individual resistances. In other words, [math]\displaystyle{ {\frac{1}{R_{equivalent}}} = {\frac{1}{R_1}} + {\frac{1}{R_2}} + {\frac{1}{R_3}} + ... + {\frac{1}{R_n}} }[/math] for n resistors in parallel.

Because 1/R = (σA)/L, if every resistor is composed of the same material and has the same length, [math]\displaystyle{ A_{equivalent} = A_1 + A_2 + A_3 + ... + A_n }[/math]

Examples

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